De Rham cohomology and integration in manifolds (Q2192176)

The aim of this paper is to present a cohomological approach to the definition of the integral of a differential form on smooth compact connected orientable manifolds, with or without boundary, based on de Rham cohomology and not using any measure theory. The approach here consists in supplying the manifold with additional structures: an orientation, a fixed finite atlas consisting of positive charts, and a fixed partition of unity subjected to the atlas. In other words, the author shows that de Rham cohomology can be used to define and compute the integral of a volume form on an oriented compact manifold by supplying the latter with a finite atlas and a partition of unity subjected to this atlas without appealing to any measure theory. This paper is organized as follows: In Section 1, the author presents three simple examples illustrating his constructions and results. In Section 2, he shows how the integral can be defined for prepared manifolds. In Section 3, the author considers the particular case of dimension 2 (surfaces) and proves that in that case a canonical Riemann metric can be defined on the prepared surface, and his definition of the integral yields the same result as the usual integral w.r.t. that metric. Section 4 is devoted to some speculations about the further development of his approach, in particular to situations where no measure theory is possible in principle, such as in the case of certain Feynman path integrals.